Finding total charge by integrating over a non-uniform charge density

Estimated Time: 30 minutes

Student groups are assigned a particular charge density that varies in space and ask to calculate the total charge.

1. Integration (particularly making “u subsitutions”)
2. Conceptual understanding of charge density

• Tabletop Whiteboard with markers
• A handout for each student

We usually start with a mini-lecture reminder that total charge is calculated by integrating over the charge density. We start the activity with the formulas $Q=\int\rho(r')d\tau'$, $Q=\int\sigma(r')dA'$, and $Q=\int\lambda(r')ds'$ written on the board.

This activity helps students practice the mechanics of making total charge calculations.

• Area and Volume Elements Students have some difficulty with the differential area and volume elements in cylindrical and spherical coordinates. We usually do the * Integrating in Curvilinear Coordinates:Finding dA and dV activity before this one, but students still need additional practice with this.
• Order of Integration When doing multiple integrals, students rarely think about the geometric interpretation of the order of integration. If they do the $r$ integral first, then the are integrating along radial line. What about $\theta$ and $\phi$. If this topic does not come up in the small groups, it makes a rich discussion in the wrap-up.
• Limits of Integration some students need some practice determining the limits of the integral. This issue becomes especially important for the groups working with a cylinder - the handout does not give the students a height of the cylinder. There are two acceptable resolutions to this situation. Students can “name the thing they don't know” and leave the height as a parameter of the problem. Students can also give the answer as the total charge per unit length - turning the volume charge density into a linear charge density. We usually talk the groups through both of these options.
• Dimensions Students have some trouble determining the dimensions of constants. Making students talk through their reasoning is an excellent exercise. In particular, they should know that the argument of the exponential function must be dimensionless.
• Integration Some students need a refresher in integrating exponentials and making substitutions.

You may ask two groups to present their solutions, one spherical and one cylindrical so that everyone can see an example of both. Examples (b) and (f) are nice illustrative examples.

You may want to augment this activity by having students find total charges from surface or linear charge densities.

You may also want to augment this activity with a homework problem where the order of integration matters and $r$ is not first.

This activity is included within a sequence of activities addressing Gauss’s law in integral form. The following activities are part of this sequence and can be used as preparation or extension to this activity.

• Preceding activity:
  • Acting Out Charge Densities: A kinesthetic activity in which students act as individual charges and move about the classroom to demonstrate linear, surface, and volume charge distributions.

• Follow-up activities:
  • The Geometry of Flux Sequence: This sequence of activities addresses the geometry of flux as well as allowing students ample practice in the mathematics which is used to calculate flux through various surfaces.
  • Gauss's Law--the integral version: This lecture introduces Gauss's law in integral form and serves as an introduction to the Gauss's Law activity.
  • Gauss's Law: This small group activity is the final activity in the Geometry of Flux sequence where students calculate the electric field due to various charge densities in spherical and cylindrical shells.

This is the final activity within a sequence of activities addressing Scalar Integration in Curvilinear Coordinates. The following activities are included within this sequence:

• Preceding activities:
  • Internal Energy of Derivative Machine: This small group activity uses a modified Partial Derivative Machine to measure the internal energy of a nonlinear, one dimensional system while emphasizing integration as an experimentally measurable quantity.
  • Curvilinear Coordinates: This lecture introduces students to curvilinear coordinates and highlights the notation difference of $\theta$ and $\phi$ in physics and mathematics.
  • Scalar Distance, Area, and Volume Elements: In this small group activity students derive expressions for infinitesimal distances in order to find area and volume elements in cylindrical and spherical coordinates and can be done with Pineapples and Pumpkins to give students a three dimensional object to explore the geometry and construction of a volume element.
  • Pineapples and Pumpkins: This activity can be done in small groups or as an instructor led whole class activity where a pineapple (for cylindrical) and/or pumpkin (for spherical) can be cut to demonstrate the geometry of an infinitesimal volume element used in integration.
  • Acting Out Charge Densities: This kinesthetic activity provides students with an embodied understanding of charge density and total charge by using their bodies to represent charges and act out linear, surface, and volume charge densities which prompts a whole class discussion on the meaning of constant charge density, the geometric differences between linear, surface, and volume charge densities, and what is “linear” about linear charge density.